integral element

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integral element

Wikipedia

Noun

integral element (plural integral elements)

1. (algebra, commutative algebra, ring theory) Given a commutative unital ring R with extension ring S (i.e., that is a subring of S), any element s ∈ S that is a root of some monic polynomial with coefficients in R.
   • 1956, Unnamed translator, D. K Faddeev, Simple Algebras Over a Field of Algebraic Functions of One Variable, in Five Papers on Logic Algebra, and Number Theory, American Mathematical Society Translations, Series 2, Volume 3, page 21,

     A subring of ${\displaystyle {\mathfrak {B}}}$ containing the ring ${\displaystyle o}$ of integral elements of the field ${\displaystyle k_{0}(\pi )}$, distinct from ${\displaystyle {\mathfrak {B}}}$, and not contained in any other subring of ${\displaystyle {\mathfrak {B}}}$ distinct from ${\displaystyle {\mathfrak {B}}}$, is called a maximal ring of the algebra ${\displaystyle {\mathfrak {B}}}$. In a division algebra, the only maximal ring is the ring of integral elements.

   • 1970 , John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, 2003 Softcover Reprint, page 172,

     If ${\displaystyle {\mathfrak {S}}}$ is the ring of integral elements in a commutative ring ${\displaystyle {\mathfrak {T}}}$ (over a subring ${\displaystyle {\mathfrak {R}}}$) and if the element ${\displaystyle t}$ of ${\displaystyle {\mathfrak {T}}}$ is integral over ${\displaystyle {\mathfrak {S}}}$, then ${\displaystyle t}$ is also integral over ${\displaystyle {\mathfrak {R}}}$ (that is, contained in ${\displaystyle {\mathfrak {S}}}$).

   • 2004, Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko, Algebras, Rings and Modules, Volume 1, Kluwer Academic Publishers, page 209:

     In this paper the notion of the ring of all integral elements of a number field was put in the central place of his^{[Richard Dedekind's]} theory.

Usage notes

• Element ${\displaystyle s}$ is said to be integral over ${\displaystyle R}$.
• The ring ${\displaystyle S}$ is also said to be integral over ${\displaystyle R}$, and to be an integral extension of ${\displaystyle R}$.
• The set of elements of ${\displaystyle S}$ that are integral over ${\displaystyle R}$ is called the integral closure of ${\displaystyle R}$ in ${\displaystyle S}$. It is a subring of ${\displaystyle S}$ containing ${\displaystyle R}$.
• If ${\displaystyle R}$ and ${\displaystyle S}$ are fields, then ${\displaystyle s}$ is called an algebraic element and the terms integral over and integral extension are replaced by algebraic over and algebraic extension (since the root of any polynomial is the root of a monic polynomial).

Translations

See also

• algebraic integer
• algebraic number
• integral closure
• integral extension

Further reading

• Algebraic element on Wikipedia.Wikipedia
• Integral closure of an ideal on Wikipedia.Wikipedia
• Integral extension of a ring on Encyclopedia of Mathematics
• Algebraic extension on Encyclopedia of Mathematics
• Integral Extension on Wolfram MathWorld
• Algebraic Extension on Wolfram MathWorld